MathDB

6

Part of 2011 Saint Petersburg Mathematical Olympiad

Problems(3)

Interesting point

Source: St Petersburg Olympiad 2011, Grade 11, P6

9/15/2017
ABCDABCD - convex quadrilateral. MM -midpoint ACAC and ∠MCB=∠CMD=∠MBA=∠MBC−∠MDC\angle MCB=\angle CMD =\angle MBA=\angle MBC-\angle MDC. Prove, that AD=DC+ABAD=DC+AB
geometry
Lights in garland

Source: St Petersburg Olympiad 2011, Grade 10, P6

9/15/2017
We have garland with nn lights. Some lights are on, some are off. In one move we can take some turned on light (only turned on) and turn off it and also change state of neigbour lights. We want to turn off all lights after some moves.. For what nn is it always possible?
combinatorics
Sequence and divisors

Source: St Petersburg Olympiad 2011, Grade 9, P6

9/15/2017
There is infinite sequence of composite numbers a1,a2,...,a_1,a_2,..., where an+1=an−pn+anpna_{n+1}=a_n-p_n+\frac{a_n}{p_n} ; pnp_n is smallest prime divisor of ana_n. It is known, that 37∣an37|a_n for every nn. Find possible values of a1a_1
number theory